1. Field of the Invention
The present invention relates to a method of generating a hybrid grid of a heterogeneous formation crossed by one or more geometric discontinuities in order for example to carry out simulations.
The method is more particularly applied to formation of a grid suited to an underground reservoir crossed by one or more wells, or by fractures or faults, in order to model displacements of fluids such as hydrocarbons.
2. Description of the Prior Art
Grid generation is a crucial element for the new generation of reservoir simulators. Grids allow to describe the geometry of the geologic structure studied by means of a representation in discrete elements wherein simulation is performed according to a suitable numerical pattern. Better comprehension of physical phenomena requires 3D simulation of the multiphase flows in increasingly complex geologic structures, in the vicinity of several types of singularities such as stratifications, faults, pinchouts, channels and complex wells. All this complexity has to be taken into account first by the grid which has to reproduce as accurately as possible the geologic information in its heterogeneous nature.
Grid modelling has made great advances during the past few years in other fields such as aeronautics, combustion in engines, structure mechanics, etc. However, the gridding techniques used in the other fields cannot be applied to petroleum applications because the professional constraints are not the same. For example, in reservoir simulation, the numerical patterns are constructed from control volumes in order to better respect the mass conservation in the case of transport equations of hyperbolic nature. The grid must be a “block-centered” type grid, that is the nodes must be inside each layer and the boundaries of each block must follow the interface between the layers. Now, if this constraint was not taken into account, the nodes would naturally be placed along the faults and along the stratification boundaries. The consequence of this would be that these interfaces would pass through the control volume used. The saturation, constant in the control volume, could not consider discontinuities and the results would not be accurate. It is therefore necessary to develop new techniques that are better suited to petroleum application requirements.
Cartesian grids, which are commonly used in current commercial simulators, are unsuited for solving these new problems posed by the development of petroleum reservoirs. Cartesian grids, based on parallelepipedic elements, do not allow representation of such complex geometries.
There is a well-known method of generating structured 3D hexahedral grids of CPG (Corner-Point-Geometry) type which respects the geometry of the bodies. It is described in French patent 2,747,490 U.S. Pat. No. 5,844,564) filed by the assignee and also in the following publication:
Bennis Ch. Et al. “One More Step in Gocad Stratigraphic Grid Generation: Taking into Account Faults and Pinchouts”; SPE 35526, Stavanger, 1996.
This grid type is more flexible than cartesian grids because it consists of any hexahedral elements that can be degenerated. It strictly considers the horizons, the faults and it allows representation of certain unconformities such as pinchouts because its construction is based on these elements. However, this type of grid does not allow solution of all geometric complexities such as, for example, circular radial grids around complex wells. It is possible to form separately the grid of the reservoir and the grids around the wells but it is difficult to represent several objects in the same CPG type reservoir grid because of connection problems linked with the structured nature of the grid.
Another approach is also known where 3D grids are only based on tetrahedral Delaunay elements, with a circular radial refinement around the wells, being automatically generated. The advantage of such an approach is that it is entirely automatic and does not practically require the user's attention. However, this method has drawbacks which make the results obtained difficult to use:
there are on average five times as many grid cells as in a CPG type grid for the same structure, which is very disadvantageous for simulation calculations,
unlike the structured grids which are easy to display, to explore from the inside and to locally modify interactively, it is very difficult and sometimes impossible to properly control the tetrahedral grids because of their size and especially because of their non-structured nature. This poses problems for validating the grid from a geometric point of view as well as for understanding and validating the result of a simulation on this type of grid.
Other approaches are also well-known, which allow generation of grids, notably grids based on control volumes generated from a triangulation associated with techniques of aggregation of the triangles (or tetrahedrons) into quadrangles allowing the number of grid cells to be reduced. Although promising results were obtained with these new grids, precise representation of the geologic complexity of reservoirs and wells remains a subject for research and development. Despite their hybrid nature, they remain entirely unstructured and would therefore be very difficult to manage and to handle in real 3D. Furthermore, taking account of real 3D faults and deviated wells would greatly increase this difficulty.
French patent application 99/15,120, filed by the assignee, describes a method of generating a 3D hybrid grid on a heterogeneous medium, comprising using a first structured grid, of CPG type for example, and structured radial grids around well or pipe trajectories drawn or imported into the model in order to better take into account the particular constraints linked with the flows in the vicinity of these wells. These grids are combined by including the radial grids around around each well in the global reservoir grid after forming therein cavities that are large enough to allow formation of unstructured transition grids.
There are various well-known techniques for forming unstructured grids. These techniques can be based on canonical polyhedrons (tetrahedrons, pentahedrons, pyramids, etc.) according to the numerical pattern being used, and entirely 3D solutions applicable to these grid types are known.
The method according to the invention is intended for 2.5D generation of transition grids based on Voronoi type or similar polyhedrons in order to apply control volume type numerical patterns, which are reduced to a 2D problem provided that all the layers of the well grids and of the reservoir grid can be projected vertically in a horizontal plane so as to form identical grid cells.
Such a grid must meet certain conditions:
the segment connecting the centers or sites of two neighboring cells must be orthogonal to the side common to the two cells,
two neighboring cells must entirely share the side that connects them, that is a cell can 5 be connected to only one other cell by the same side, and
the cells must be convex.
In 2D, these constraints impose that:
the edges of the boundary polygons must not be modified (each one must correspond to an edge of a cell created, without subdivision of the edge), and the cells which are created must be convex,
each cell must have a center such that the straight line connecting the centers of two adjacent cells is perpendicular to the edge shared by the two cells, and
the centers of the cells must be inside their cell.
Other entirely non-structured approaches are also well-known, which allow generation of grids, notably grids based on control volumes generated from a triangulation, associated with techniques of aggregation of the triangles (or tetrahedrons) into regular quadrangles allowing the number of grid cells to be reduced.
Voronoi diagrams are formed, as it is well-known, by determining around a set of sites P distributed in a space regions consisting of series of points which are closer to each site than to any other site P of the set. This construction technique is described in detail notably in the following documents:
Aurenhammer F., 1991: Voronoi “Diagrams: A Survey of Fundamental Geometric Data Structure,” ACM Comput. Sun., 23, 345–405,
Fortune S., 1992: Voronoi Diagrams and Delaunay Triangulations, pages 225–265 of D. Z. Du & F. K. Hwang (eds), Computing in Euclidean Geometry, 2nd edn., Lecture Notes Series on Computing, vol. 4, Singapore, World Scientific.
The cells of a Voronoi diagram meet the following constraints: the orthogonality between the centers and the adjacent edge is respected and the cells are convex. It is therefore natural to propose filling the cavity with Voronoi cells while trying to consider the edge geometric constraints. However, if a Voronoi diagram is created from the existing points (ends of the polygon edges) (see FIG. 6), it can be observed that this diagram does not meet the geometric constraints, that the polygon edges do not belong to the diagram and that the edges are cut.
In order to introduce the edges of the boundary polygons, it is possible to duplicate the existing points. The Voronoi diagram thus has to pass through the edges. However, this is not yet sufficient because, although the edges are in the diagram, they still are divided (see FIG. 7).
Since the edges of the Voronoi diagram correspond to the mid-perpendiculars of the segments defined by two sites, passing through these mid-perpendiculars through the ends of the edges defined by the polygons can be tried. For a given end point, the sites of the Voronoi diagram are therefore selected along edges adjacent to this point, equidistant therefrom. Since the points are duplicated, one can be certain to obtain an edge of the Voronoi diagram on the edge of the polygon. However, if the result obtained is correct in the case of a regular polygon (FIG. 8a), this is not the case for a non-regular polygon (FIG. 8b). When the last sites are placed, they are not necessarily at the same distance from the point as the first placed sites.
Generally speaking, using Voronoi cells for filling the cavity does therefore not seem to be suitable.